Geometry formula sheet

Units: P, C: units · A, LA, SA: units² · V: units³. Use the same unit for all positive lengths. SA: closed surface · LA: side only. π exact · ≈ rounded.

2D · Perimeter & area

Rectangle / square

Area: $A = L W$
Perimeter: $P = 2 (L + W)$
Square area: $A = a^{2}$
Square perimeter: $P = 4 a$

L length · W width · a square side

Four right angles. Square: L = W = a. All lengths > 0.

Area uses square units; perimeter uses length units.

Triangle

Area: $A = \frac{b h}{2}$
Perimeter: $P = b + c + e$

b base · c, e other sides · h perpendicular height

h > 0; its foot may lie on the extended base. Each side < the sum of the other two.

A sloping side is not h. Base and height alone do not determine perimeter.

Circle

Radius / diameter: $d = 2 r$
Circumference: $C = 2 π r = π d$
Area: $A = π r^{2} = \frac{π d^{2}}{4}$

r radius · d diameter · O center

r > 0. π is exact; use ≈ for rounded answers.

Using d as r makes the area four times too large.

Trapezoid

Area: $A = \frac{(b_{1} + b_{2}) h}{2}$
Perimeter: $P = b_{1} + b_{2} + c + e$

b₁, b₂ parallel bases · c, e legs · h perpendicular height

Bases are parallel; h > 0 is their perpendicular distance.

Use h, not a sloping leg. Perimeter needs all four sides.

3D · Surface area & volume

Rectangular prism / cube

Volume: $V = L W h$
Surface area: $\begin{array}{l} SA = 2 L W \\ + 2 L h + 2 W h \end{array}$
Cube volume: $V = a^{3}$
Cube surface area: $SA = 6 a^{2}$

L length · W width · h height · a cube side

Closed, six faces. Cube: L = W = h = a. All lengths > 0.

Surface area counts all six faces; it is not material volume.

Right circular cylinder

Volume: $V = π r^{2} h$
Lateral area: $LA = 2 π r h$
Closed surface area: $SA = 2 π r^{2} + 2 π r h$
Open-top area: $A = π r^{2} + 2 π r h$

r base radius · h perpendicular height · LA curved side only

Right circular cylinder; r, h > 0. No wall thickness.

Closed: two bases. Open top: one base. Lateral: no bases.

Right circular cone

Volume: $V = \frac{π r^{2} h}{3}$
Lateral area: $LA = π r s$
Closed surface area: $SA = π r^{2} + π r s$
Slant height: $s^{2} = r^{2} + h^{2}$

r base radius · h perpendicular height · s slant height

Right circular cone only; s > r > 0, h > 0.

Volume uses h, not s. Closed area includes the base.

Sphere

Radius / diameter: $d = 2 r$
Volume: $V = \frac{4 π r^{3}}{3}$
Surface area: $SA = 4 π r^{2}$

r radius · d diameter · O center

Complete sphere; r > 0. No hemisphere or shell.

Surface area uses square units; volume uses cubic units.

Four measurement mistakes to avoid

Diameter is not radius

Wrong: d = 10 cm → A = 100π cm²

Use: r = 5 cm → A = 25π ≈ 78.54 cm²; C = 10π ≈ 31.42 cm

Halve the diameter first. Using 10 as the radius makes the area four times too large.

Use the perpendicular height

Wrong: Sides 8, 5, 5 cm → A = 8 × 5 ÷ 2 = 20 cm²

Use: h = 3 cm → A = 12 cm²; P = 8 + 5 + 5 = 18 cm

The height splits the 8 cm base into two 4 cm halves. Each half forms a 3–4–5 right triangle.

Cone volume uses h

Wrong: r = 3 cm, h = 4 cm, s = 5 cm → V = 15π cm³

Use: V = π × 3² × 4 ÷ 3 = 12π ≈ 37.70 cm³

Using slant height 5 instead of perpendicular height 4 overstates the volume by 25%.

Count the cylinder bases

Wrong: r = 3 cm, h = 5 cm → closed SA = 30π cm²

Use: LA = 30π cm²; closed SA = 48π cm²; open-top area = 39π cm²

The curved side is 30π cm². Add two 9π cm² bases for a closed cylinder, or one for an open top.